Let \({\cal M}\)  be a general manifold. The tangent space of the manifold \({\cal M}\)  at \(x\)  is denoted by \(T_x{\cal M}\)  and the tangent bundle is denoted by \(T{\cal M}\) .
A retraction [1]} is a smooth mapping from the tangent bundle to the manifold such that, for all \(x\in {\cal M}\) , \(\eta _x\in T_x{\cal M}\) , (i) \(R_{x}\left(0_{x}\right)= x\)  where \(0_{x}\)  denotes the the zero element of \(T_{x}{\cal M}\) , and (ii) \(\frac{d}{dt}R_{x}\left(t\eta _{x}\right)\vert _{t = 0} = \eta _{x}\) .
By [1]}, a retraction on the quotient manifold \(\overline{{\cal M}}_{\mathbf {r}}/{\cal G}\)  is given by
\(R^{Q}_{\left[\bar{{\cal X}}\right]}\left(\xi _{\left[\bar{{\cal X}}\right]}\right)= \pi \left(R^{Q}_{\bar{{\cal X}}}\left(\bar{\xi }_{\bar{{\cal X}}}\right)\right)=\left[\bar{R}^{Q}_{\bar{{\cal X}}}\left(\bar{\xi }_{\bar{{\cal X}}}\right)\right],\) 
